The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
topology," the language used in these quotes gets the message
across without fancy metaphors or what-have-you. That's probably the
style of most mathematicians.

Occasionally, however, one is surprised by unexpectedly colorful
language in a mathematics paper. If I remember correctly, a paper of
Gerald Sacks once described a distinction as being

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56

Latest paper, my co-author put in "but we will choose a more painful way, because there is nothing like pain for feeling alive" but the referee jumped on it.
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Will JagyApr 23 '10 at 5:09

16

Maybe I should expand the question to include colorful language cut from serious mathematics papers :)
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John StillwellApr 23 '10 at 5:18

32

By the way, your remark reminds me of another in a similar spirit that made it into the Princeton Companion. In his article on algebraic geometry, János Kollár says of stacks: "Their study is strongly recommended to people who would have been flagellants in earlier times."
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John StillwellApr 23 '10 at 7:49

27

I was actually rather surprised recently by a referee who did not know the phrase “red herring”, and had to look it up. He insisted that we change it to something more understandable. It makes me wonder how much “colourful” language is weeded out by referees, and whether the mathematical literature is poorer for it.
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Harald Hanche-OlsenApr 24 '10 at 2:31

25

@Harald: If you intend your mathematical papers to be read by a wide range of readers, then write them in simple language, suitable for those who are relative beginners in English. I remember reading long ago some metaphoric phrase in a mathematics research paper, then imagining students all over the world getting out their English dictionaries, looking it up, and still not understanding what it meant. (I no longer remember what the phrase was, just this reaction to it.)
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Gerald EdgarApr 24 '10 at 15:43

There is the famous (and with contradictory interpretations) cry from Jean Dieudonné "à bas Euclide !", "Down with Euclide !". His books and prefaces are good sources for strong (and dated) opinions on what was "good" or "productive" mathematics and what was not.

We give a baseball interpretation of the partial order $\preceq$. The $m$th symbol in a word $w$ is the $m$th inning. The sum of the first $m$ symbols is the score after $m$ innings. The relation ${w_1\preceq w_2}$ means precisely that after every inning, ${w_1}$ is not losing.
(Note that this is low-scoring baseball: every inning, each team scores $\pm1$ run. It is also fixed: the end result is tied. The lead changes precisely when words are not comparable; comparable words are uninteresting as spectator sport. Two words are comparable if and only if they describe a low-scoring, fixed, and uninteresting baseball game.)

Later in the paper, there is proof by skiing (with comparably colourful language) and various bypass shennanigans.

Two that I like can be found on p. 756 of Edgar R. Lorch's Amer. Math. Monthly paper "Continuity and Baire functions" (Volume 78, 1971, pp. 748-762):

[...] the reader is reminded of the fact that sets which are of type F_sigma_delta_sigma or G_delta_sigma_delta and not of lower type--with respect to any of the classic topologies--are very thinly scattered through the literature. In fact, looking for them is almost like hunting for unicorns.

In order to penetrate further into this subject it is necessary to give an appropriate structure to T, the set of all coherent topologies. As mentioned earlier, this appropriate structure is itself a topology. This circumstance, that a collection of topologies is topologized, may seem a bit incestuous.

Entertaining (and I'm sure we all know books like that in our respective fields)... but aren't we looking for instances of such language in serious math(s) papers, the point being to find levity defying gravity?
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Yemon ChoiMar 11 '11 at 1:10

In T.Y.Lams book "Lectures on modules and rings" there is a chapter on quotient rings. The three subsections of which are named "The Good", "The Bad" and - of course - "The Ugly". The three subsections are about existence and uniqueness of a "localization" with the universal property in the noncommutative case ("The Good" though nothing is good about this localization in general, everything nice is lost in the general case), Mal'cev's example of a domain that cannot be embedded into a division ring ("The Bad") and further theorems about which classes of rings can be embedded together with example that there need not to be a unique minimal such division ring ("The Ugly").

No-one seems to have mentioned Joe Diestel (although "colorful" is maybe the wrong word-- perhaps because of my English interpretation of what this means-- but "lighthearted" is correct). For example, "Sequences and Series in Banach Spaces" we have the section on "Mathematical Sociology" when introducing Ramsey Theory (to talk about one set "accepting" or "rejecting" another). It's hard to pick out any particular quote, but the whole book is somehow far more lively and informal (without, somehow, even managing to be less than 100% accurate) than most maths books.

Within the menagerie of objects studied in contemporary probability theory, there are a number of related "animals" that have attracted great interest amongst probabilists and physicists in recent years.

One of my favorites has always been Hermann Weyl's "... the gods have imposed upon my writing the yoke of a foreign language that was not sung at my cradle" (in the preface to his classic text `The Classical Groups: their Invariants and Representations') to excuse his supposedly poor English. This was a conceit of course---as the quote itself shows his command of English was impeccable.

@Hans: Really? anthropomorphize? When I look at how many "monsters", "beast" etc. are out there, then I tend to think that at least the "official" termininology is more animalistic.
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Johannes HahnJan 20 '11 at 10:56

2

I once followed a lecture of David Goss where he started calling his objects "guy", passed on to something like "unpleasant fellow" (when he was revealing some undesired properties of that object) and ended up calling it "sucker" - repeatedly and emotionally.
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Peter ArndtOct 22 '11 at 20:12